Migration Modelling forMultilayer Materials · MLM made of 2 to 10 homogenous Layers ρi = const. -...

EURL NRL FCM Working Groups, Ispra/Italy, 14-16.04.2010

Migration Modelling for Multilayer

Materials

Migration Modelling for Multilayer

Materials

Peter MerceaFABESFABES GmbH - Schragenhofstraße 35, 80992 Munich / Germany


Topics :

�Basic aspects of modelling migration from multilayer materials

�Finite Difference (FD) Method to solve the Mass Transport Equation

�Performances and Validation of the FD NUMERIC Algorithm

�Input Menus and Results computed by MIGRATEST©EXP

�Case Examples: - „Invisible set-off“

- Functional barrier


CFtCCFtFt

Layer 1

Food

Layer 2

Layer 3

Multi-Layer Packaging

To be compared with the legal migration limits !

Migration from a Multi-layer Packging into a Food

MigrantCpo (mg/kg)


Basic Aspects of Migration from Multi-Layer Materials (MLM)


When k >2 No analytical solution of the Mass

Transport Equation

To solve Fick‘s Equation one can use:

Numerical methods

Variational methods

Monte-Carlo methods

Finite Element Analysis (FEA)

Finite Volume Method (FVM)

Boundary Element Method (BEM)

Finite Difference Method (FDM)

SFPP3-INRA

2

2

x

cD

t

c kk

k

∂

∂=

∂

∂Fick‘s

2nd Equation

Mass Transport Equation in a Multi-Layer in contact with a Food1.

Basic Aspects of Migration from Multi-Layer Materials (MLM)


Discretisation scheme

FD Discretisation scheme

only the boundary is discretised

Numerical Methods to Solve a Differential Equation over a Domain

Finite Element Method, FEM

Finite Volume Method, FVM

Boundary Element Method, BEM

Finite Difference Method, FD

SFPP3-INRA


Constant mesh:

each Layer divided in equally thick slabs

hi = dpi /M

M=10

Finite Differences Method to Calculate Migration from Multi-Layers in Food


Adaptive mesh:

each Layer divided in M slabs of varying thickness.Higher mesh density close to the boundaries.M=22



V.Tosa, K. Kovacs, P. Mercea and O. Piringer,

2008 AIP Conference Proceedings, 1048, 802.

- Write equations to approximate the derivatives of the

diffusion equation, at each mesh point, with appropriate

discrete formulas.

- Solve the equations and calculate the migrant flux.

2

2

x

cD

t

c kk

k

∂

∂=

∂

∂



Expert Software for the Calculation of Diffusion/Migration from a Multilayer Material into a Contact Medium

Dr. Valer Tosa © 2006 updated 2008, 2009 & 2010

One-dimension - Plane Geometry

FD Method - Relatively simple and suited to quantify the migration

from MLM’s



- free choice of diffusion coefficients Dpi in the MLM and DF in the Contact Medium

- partitioning, Kpi,pk at each interface i,k of the MLM – Contact Medium system

ρiMLM made of 2 to 10 homogenous Layers = const.

- constant diffusion coefficients in each layer Dpi = const.

Solving the Diffusion/Migration Equations for:

liquid, viscous or solid

ρF = const.

Contact Medium

homogenous

one sided or two sidedContact between the MLM and Medium

dpi = const. - constant thickness

Main Features of the Expert Software NUMERIC


Migrant concentration profiles in each layer of the MLM and in the Contact Medium

Main results computed by:


Main results computed by:

Time dependent mean concentrations in each layer of the MLM and in the Contact Medium


Performances of the Finite Difference (FD) algorithm:

Solves in MIGRATEST©EXP the

diffusion/migration equations for:

- a single Migrant with Molecular Weight, Mw, from 2 to 3,500 g/mol

- Migrant initially distributed in one, some or all Layers of the MLM

- initial concentration/s, C p,o‘s, of Migrant from 1 to 500,000 mg/kg

- thickness, dp, of each Layer from 1 to 100,000 µm

- density, ρρρρ, of each Layer from 0.1 to 20 g/cm³


- Migrant diffusion coefficients, Dp‘s, defined for each Layer- D‘s, from 10-2 cm²/s (very well mixed liquid) to 10-25 cm²/s (diffusion in solids)

- Migrant partitioning coefficients between Layers, Kpi,pj

- Kpi,pj‘s from 0.01 (extraction) to 1,000,000 (virtually a barrier)

Solves in MIGRATEST©EXP the

diffusion/migration equations for:

Performances of the Finite Difference (FD) algorithm:

- Migrant partitioning coefficients between MLM and Food, Kp,f - Kp,f from 0.01 (extraction) to 1,000,000 (virtually a barrier)


Black BoxWarning for non-experts:

Do not open !!!

Diffusion Coeffs.,Dpi and DF

Partition Coeffs.,KPiPj and KPF

INPUT DATA

Contact area, A

Volume of food, VF

Thicknesses, dpi

Densities, ρρρρP and ρρρρF

Migration time, t

Initial conc., Cpo

START

RESULTS

Are the calculations with NUMERIC correct or not ? This is the question !


Validation of the Finite Difference (FD) algorithm:

Validation of the FD algorithm isnecessary in order to assure the

user of the software MIGRATEST©EXP that the results

computed are correct:

- Validation by comparison with data calculated with other methods oralgorithms

- Validation by mathematical analysis of the computed results

- Validation by comparison between input data and calculated results


Test 1 Comparison with results calculated with analytical algorithms

ρi = ρj = const Di = Dj = const

all ki,j, = kjF = 1

DF = 10-5 cm²/s - liquid medium/food

Cp,i (0) = Cp,j (0) = 1000 mg/kg

CF (0) = 0 mg/kg


CF,t = 8.9621 mg/kg

Relative difference of less than 0.1% between the numerical and analytical results !

Test 1 Comparison with results calculated with an analytical algorithm

CF,t = 8.962 mg/kg


Comparison with results calculated with an analytical algorithm


Test 2 Continuity of concentration profiles at interfaces

ρi = ρj = const

all ki,j, = kjF = 1

DF = 10-5 cm²/s - liquid medium/food

Cp,1 (0) = Cp,2 (0) = Cp,4 (0) = Cp,5 (0) = 0 mg/kg

CF (0) = 0 mg/kg

Di = Dj

Cp,3 (0) = 1000 mg/kg


Test 2 Continuity of concentration profiles at interfaces

Relative error

less that 0.5%

for a mesh of

200 or larger


Test 3 Values of migrant concentration jump at interfaces versus input partition coefficient

Partition values defined as input values

at each interface

all Ki,j, and KjF = 1

Di = Dj

Concentration jumps at interface calculated by NUMERIC

Cp,3 (0) = 1000 mg/kg

Relative deviation between input and calculated data


Cp,3 (0) = 1000 mg/kg

100 200 300 4000,02

0,03

0,04

0,05

0,06

0,07

0,08

Rela

tive

De

via

tio

n (

%)

Number of mesh points

Relative deviation between the ration Ci / Cj and the partitioning Kij

Test 3 Values migrant concentration jump at interfaces versus input partition coefficient

Ki,j, = 20

Partition defined as input value

at the interface ijfrom 210 µm

Concentration values calculated by NUMERIC at interface ij and different time points:


Test 4 – Overall mass conservation in the MLM – Contact Medium System

0 10 20 300

1

2

3

4

1

0-6 δ

qr

Time (hours)

Rela

tive t

ota

l m

ass

devia

tion (m

g)

Cp,3 (0) = 1000 mg/kg

Mass conservation assumption

∑=

+=n

i

titFt qqq1

,,


Conclusions Conclusions

very good accuracy & results

fast computation with a regular PC

FD model used in

suited for plane geometry !

Development of a user-friendly PC program


Main Features & Menus of the Software:


Input Menus


Input Menus– Migration from the MLM into a Contact Medium


Input Menus


Values

of mean

concentrations

and amounts

For 12 time intervals

in each layer of the MLM

Graphs

of concentration

profiles&

mean concentrations

Results of Computations


Values

of mean

concentrations

and amounts

For 12 time intervals

in each layer of the MLM and the CM

Graphs

of concentration

profiles&

mean concentrations

Results of Computations


Layer 1

Layer 2

Layer 1

Layer 2

Contact Medium

Time dependent mean

Migrant Concentration

in each Layer

of the MLM and the CM

Time dependent mean


in each Layer


Contact MediumTime dependent mean


in each Layer


Contact MediumContact Medium

Results of Computations – 2nd Step


Case ExamplesCase Examples1. Process of „Invisible set-off“ during Manufacturing & Storage of a Multilayer

Material

Lamination or extrusion of the Multi-Layer Film

Tw

isti

ng

to a

bobbin

Zoom in

The Multi-Layer Material manufactured as a bobbin and stored for a certain time.


1st Winding 2nd Winding 3rd Winding

Migrant, CpoMigrant, Cpo

Process of „Invisible set-off“ during Manufacturing & Storage of a Multilayer

Material


1st Winding 2nd Winding 3rd Winding

Migrant, CpoMigrant, Cpo

Contact between windings

Process of „Invisible set-off“ during Manufacturing & Storage of a Multilayer

Material


Input Menus


Diffusion from Layer 1

Results of Computations – without „invisible set-off“


Results of Computations – with „invisible set-off“

Diffusion from Layer 1 Contamination of Contact Layer due to migration from the next multi-layer

Diffusion from Layer 1


Results of Computations – Migration without considering the „invisible set-off“


Results of Computations – Migration with considering the „invisible set-off“


Case ExampleCase Example2. Functional Barrier in a Multi-layer Packaging Material

2. Estimation of Migration through a Functional Barrier


Input Data


Input Data - Migration from the MLM into a Food for 4 years @20 °C


Migration from the MLM into a Food for 4 years @20 °C – without PET layer


Migration from the MLM into a Food for 4 years @20 °C – with barrier effect of PET


Migration from the MLM into a Food for 30 days @20 °C –with PET barrier layer but with „invisible set-off“ from the printing ink layer

Migration Modelling forMultilayer Materials · MLM made of 2 to 10 homogenous Layers ρi = const. -...

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